câu10

thế là đc rồi .

bạn học giỏi đó, nhưng mong đọc lại nội quy

\(tanx=tan\frac{3\pi}{11}\Rightarrow x=\frac{3\pi}{11}+k2\pi\)

Do \(\frac{\pi}{4}\le x\le2\pi\)

\(\Rightarrow\frac{\pi}{4}\le\frac{3\pi}{11}+k2\pi\le2\pi\)

\(\Rightarrow-\frac{1}{88}\le k\le\frac{19}{22}\)

Mà \(k\in Z\Rightarrow k=0\)

Vậy pt có đúng 1 nghiệm trên đoạn đã cho

3.3 d)

\(\sin8x-\cos6x=\sqrt{3}\left(\sin6x+\cos8x\right)\\ \Leftrightarrow\sin8x-\sqrt{3}\cos8x=\sqrt{3}\sin6x+\cos6x\\ \Leftrightarrow\sin\left(8x-\dfrac{\pi}{3}\right)=\sin\left(6x+\dfrac{\pi}{6}\right)\\ \Leftrightarrow\left[{}\begin{matrix}8x-\dfrac{\pi}{3}=6x+\dfrac{\pi}{6}+k2\pi\\8x-\dfrac{\pi}{3}=\pi-\left(6x+\dfrac{\pi}{6}\right)+k2\pi\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{12}+k\dfrac{\pi}{7}\end{matrix}\right.\)

3.4 a)

\(2sin\left(x+\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \Leftrightarrow2cos\left(\dfrac{\pi}{2}-x-\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \Leftrightarrow2cos\left(-x+\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \Leftrightarrow2cos\left(x-\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \)

Chia hai vế cho \(\sqrt{2^2+4^2}=2\sqrt{5}\)

Ta được:

\(\dfrac{1}{\sqrt{5}}cos\left(x-\dfrac{\pi}{4}\right)+\dfrac{2}{\sqrt{5}}sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3}{4}\\ \)

Gọi \(\alpha\) là góc có \(cos\alpha=\dfrac{1}{\sqrt{5}}\)và \(sin\alpha=\dfrac{2}{\sqrt{5}}\)

Phương trình tương đương:

\(cos\left(x-\dfrac{\pi}{4}-\alpha\right)=\dfrac{3}{4}\\ \Leftrightarrow x=\pm arscos\left(\dfrac{3}{4}\right)+\dfrac{\pi}{4}+\alpha+k2\pi\)

Xét \(x\ne1\)

\(\left(1+x+...+x^{10}\right)^{11}=a_0+a_1x+...+a_{110}x^{110}\)

\(\Leftrightarrow\left(x-1\right)^{11}\left(1+x+...+x^{10}\right)^{11}=\left(x-1\right)^{11}\left(a_1+a_1x+...+a_{110}x^{110}\right)\)

\(\Leftrightarrow\left(x^{11}-1\right)^{11}=\left(x-1\right)^{11}\left(a_0+a_1x+...+a_{110}x^{110}\right)\)

\(VP=\left(x-1\right)^{11}\left(a_0+a_1x+...\right)=\left(\sum\limits^{11}_{k=0}C_{11}^kx^k\left(-1\right)^{11-k}\right)\left(a_0+a_1x+...\right)\) (1)

Ta thấy tổng các hệ số của \(x^{11}\) trong khai triển (1) là:

\(C_{11}^0\left(-1\right)^{11}.a_{11}+C_{11}^1\left(-1\right)^{10}a_{10}+C_{11}^2\left(-1\right)^9a_9+...+C_{11}^{11}\left(-1\right)^0a_0\)

\(=-C_{11}^0a_{11}+C_{11}^1a_{10}-C_{11}^2a_9+...+C_{11}^{11}a_0=-T\)

\(VT=\sum\limits^{11}_{k=0}C_{11}^k\left(x^{11}\right)^k.\left(-1\right)^{11-k}\)

Hệ số của \(x^{11}\) trong khai triển trên là \(C_{11}^1\left(-1\right)^{10}=C_{11}^1=11\)

Mà \(VT=VP\Rightarrow-T=11\Rightarrow T=-11\)

\(\Leftrightarrow1-cosx-5cosx+2cos^2x-1=0\)

\(\Leftrightarrow2cos^2x-6cosx=0\)

\(\Leftrightarrow cosx\left(cosx-3\right)=0\)

\(\Leftrightarrow cosx=0\)

\(\Leftrightarrow...\)

Bài làm

a)dãy số U: \(2,7,12,...x\)

U là cấp số cộng\(\Rightarrow\left\{{}\begin{matrix}d=u_2-u_1=7-2=5\\u_1=2\end{matrix}\right.\)

\(U_n=U_1+\left(n-1\right)d\)

=> \(n=\dfrac{U_n-U_1}{d}+1=\dfrac{x-2}{5}+1=\dfrac{\left(x+3\right)}{5}\)

\(S_n=\dfrac{n\left(U_1+U_n\right)}{2}=\dfrac{\dfrac{\left(x+3\right)}{5}\left(2+x\right)}{2}=\dfrac{\left(x+3\right)\left(x+2\right)}{2.5}=245\)

\(x^2+5x+6=2450\)

\(x^2+5x-2444=0\)

\(\Delta=5^2-4.\left(-2444\right)=9801=\)99^2

\(\left\{{}\begin{matrix}x_1=\dfrac{-5-99}{2}< 0\left(loai\right)\\x_2=\dfrac{-5+99}{2}=47\end{matrix}\right.\)

Đáp số: x=47

b) Xét cấp số cộng 1, 6, 11, ..., 96. Ta có :

\(96=1+\left(n-1\right)5\Rightarrow n=20\)

Suy ra :

\(S_{20}=1+6+11+...+96=\dfrac{20\left(1+96\right)}{2}=970\)

và \(2x.20+970=1010\)

Từ đó : \(x=1\)

a/

\(\Leftrightarrow2sinx.cosx+2\sqrt{3}cos^2x=\sqrt{3}-2sin5x\)

\(\Leftrightarrow sin2x+\sqrt{3}\left(cos2x+1\right)=\sqrt{3}-2sin5x\)

\(\Leftrightarrow sin2x+\sqrt{3}cos2x=-2sin5x\)

\(\Leftrightarrow\frac{1}{2}sin2x+\frac{\sqrt{3}}{2}cos2x=-sin5x\)

\(\Leftrightarrow sin\left(2x+\frac{\pi}{3}\right)=sin\left(-5x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2x+\frac{\pi}{3}=-5x+k2\pi\\2x+\frac{\pi}{3}=\pi+5x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{21}+\frac{k2\pi}{7}\\x=-\frac{2\pi}{9}+\frac{k2\pi}{3}\end{matrix}\right.\)

b/

\(\Leftrightarrow sinx+\sqrt{3}cosx=2sin3x+2sinx\)

\(\Leftrightarrow sinx-\sqrt{3}cosx=-2sin3x\)

\(\Leftrightarrow\frac{1}{2}sinx-\frac{\sqrt{3}}{2}cosx=-sin3x\)

\(\Leftrightarrow sin\left(x-\frac{\pi}{3}\right)=sin\left(-3x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{\pi}{3}=-3x+k2\pi\\x-\frac{\pi}{3}=\pi+3x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{12}+\frac{k\pi}{2}\\x=-\frac{2\pi}{3}+k\pi\end{matrix}\right.\)